Problem 27 - Elementary Algebra

Question

For the following problems, use algebraic notataion. \(y+11\) divided by \(y+10,\) minus 12

Step-by-Step Solution

Answer

Answer: The simplified expression is \(\frac{-11y - 109}{y + 10}\).

Step 1: Identify the common denominator

In this case, the common denominator is \((y + 10)\), since the first term already has this denominator and the second term, 12, can be rewritten with this denominator.

Step 2: Rewrite the second term with the common denominator

We will rewrite 12 with the \((y + 10)\) as the denominator. To do this, we multiply 12 by \((y + 10)/(y + 10)\): $$12 = \frac{12(y + 10)}{y + 10}$$

Step 3: Combine the terms

Now we have both terms with the same denominator, so we can simply combine them: $$\frac{y + 11}{y + 10} - \frac{12(y + 10)}{y + 10}$$

Step 4: Simplify the expression

We now combine the terms in the numerator: $$\frac{y + 11 - 12(y + 10)}{y + 10}$$ Distribute the -12 to both terms in the parenthesis: $$\frac{y + 11 - 12y - 120}{y + 10}$$ Combine the like terms in the numerator: $$\frac{-11y - 109}{y + 10}$$ The final simplified expression is: $$\frac{-11y - 109}{y + 10}$$

Key Concepts

Understanding the Common Denominator

When dealing with algebraic fractions, finding a common denominator is crucial. It allows us to combine fractions by ensuring that the denominators are the same. Think of it like finding a common language so they can "talk" to each other.

In our exercise, the common denominator for the algebraic expressions is \((y + 10)\). The first term, \(\frac{y + 11}{y + 10}\), already has this denominator. For the constant 12, we rewrite it with \((y + 10)\) as its denominator by multiplying by \((y + 10)/(y + 10)\).

This transformation keeps the value of the original expression because multiplying by one—albeit a more complex "one" in this case—does not change the value. Once both fractions share the same denominator, we can treat the numerators as straightforward subtraction inside the same frame.

Simplifying Expressions

Simplifying expressions involves making them easier to work with without changing their value. It's like tidying up a room—making everything neat and in order.

After reorganizing the fractions with a common denominator, we need to simplify the expression further. With the fractions from our example, you start with:

• \(\frac{y + 11}{y + 10} - \frac{12(y + 10)}{y + 10}\)

Once the fractions have a common denominator, you can combine the numerators:

• \(y + 11 - 12(y + 10)\)

Then, apply distribution to simplify the terms:

• \(y + 11 - 12y - 120\)

Finally, by combining like terms (numbers that share the same variable or are constants), you get the neatly simplified form:

• \(\frac{-11y - 109}{y + 10}\)

Voila, that's your polished expression!

Algebraic Notation Basics

Algebraic notation is the way we write mathematical expressions with symbols and letters. It helps us clearly communicate complex mathematical ideas.

In our current exercise, we're using letters like \(y\) to represent variables. Variables are placeholders for numbers, which can change. The use of algebraic notation allows us to write equations or expressions that are universally understandable.

A few key components of algebraic notation include:

• **Letters/Variables**: Usually \(x\), \(y\), \(z\), etc., representing numbers.
• **Operators**: Symbols like \(+\), \(-\), \(\times\), and \(\div\) which show what mathematical operation to perform.
• **Constants**: Numbers that stay the same.
• **Parentheses**: Group terms together, dictating the order of operations.

By mastering algebraic notation, you're able to manipulate and solve expressions like the one in this exercise efficiently and universally!